Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
501 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,19)(12,17)(13,16)(14,18)(15,21)(20,23)(22,24);(1,2,3,4)(9,11,12,21,14,20,13,22)(10,15,16,19,18,24,17,23);(10,16)(11,20)(12,13)(15,19)(17,18)(23,24)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11144, #357842, #357843, … |
| maximal normal subgroup of | C2×#11144, #357842, #357843, … |
| maximal quotient of | #116997, #117035, #117041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4132);